Dα: A Fractional Spectral Framework for Number Theory and Structural Computation

We introduce a fractional spectral operator Dα, defined via the Ca- puto derivative, that bridges analytic number theory and entropy-based structural computation. The operator Dα captures memory-dependent oscillatory structures and reveals hidden symmetries in the distribution of prime numbers. When coupled with entropy-based structure indices Kα(x) = dlog H(x) dlog Dα f(x) (2), the framework detects fractional attractors and scale-invariant phase-locking within arithmetic systems. Numerical exper- iments with f (x) = x2 confirm convergence properties, with applications ranging from prime density modeling to long-memory structural inference in complex systems like aerospace structures, satellites, or deployable pan- els. This framework provides a robust spectral tool for analyzing arith- metic logic and memory-dependent systems, relevant to high-precision NASA computational architectures [5].